Preheat your oven to 350°. Spray a medium casserole dish with vegetable cooking spray, such as Pam.
Slice the squashes into lengthwise quarters, then slice crosswise. Place the squashes and carrots in a saucepan, cover with water, and salt well. Bring to a boil, reduce heat to medium, and cover. Cook for about 20 minutes, until the vegetables are very soft.
Meanwhile, in a medium skillet, melt 3 TB of the butter over medium-low heat. Stir in the onion and cook slowly until very tender and translucent, 6-8 minutes. Add the garlic and cook for another minute or so. Place the mixture into a mixing bowl.
Wipe out the skillet with a paper towel and return it to medium-low heat. Add the remaining tablespoon butter; melt. Stir in 1/2 cup of the cracker crumbs and cook for a minute or so, until crumbs are golden brown. Put them on a small plate lined with paper towels to drain.
Now drain the squash mixture. Mash vegetables slightly with a fork or potato masher. Spoon the squash into the mixing bowl with the onions. Stir in the rest of the uncooked cracker crumbs, the cheese and a dash or two of Tabasco. Salt and pepper to taste. Stir in the eggs and spoon the mixture into the prepared baking dish. Scatter the toasted crumbs over the top.
Bake the casserole, uncovered, for about 30 minutes, until golden brown and lightly firm in the center. Serve hot.